Measures of Dispersion

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May Perkins
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1 Chapter IV Meaure of Dpero R. 4.. The meaure of locato cate the geeral magtue of the ata a locate oly the cetre of a trbuto. They o ot etablh the egree of varablty or the prea out or catter of the vual tem a the evato from the average. Two trbuto of tattcal ata may be ymmetrcal a have commo arthmetc average, mea a moe. Yet wth thee pot commo they may ffer wely the catter or ther value about the meaure of locato. D. 4.. (Rage). The rage of a ugroupe efe by: R : = ma m m : mallet value the ata et ma : larget value the ata et. The rage of a groupe ata efe by: R : B p b b : lower bou of the frt cla B p : upper bou of the cla p. E. 3.. (cot.). Ugroupe. Groupe R = = 4. R = = 4.40 R. 4.. (Davatage of the Rage). It fluece by outler.. It calculato bae o two value oly: the larget a the mallet. Thu, the rage ot a very atfactory meaure of pero.

2 D. 4.. (Iterquartle Rage) The terquartle rage efe a: R Q : = E. 3.. (cot.). Ugroupe R = =.45 Q. Groupe R = =.85 Q D (Average Abolute or Lear Devato - Ugroupe). Abolute (or lear) average evato from the mea: 0.5 : = = 0.5. Abolute (or lear) average evato from the arthmetc average: : = = E. 3.. (cot.). : = 6.0 = =. : = 6. = = R (Mmum Property of the Mea) = 0.5 N =, N R D (Average Abolute or Lear Devato - Groupe) 0.5 : p 0.5 = F

3 R I cae the formato : p = F are ot avalable, they wll be replace by the terval mpot: E. 3.. (cot.) We ha: 0.5 = 6.9, = C F m m 0. 5 F m F [ 4.00, 5.00[ [ 5.00, 6.00[ [ [ [ [ Total , = D (Average Square Devato - Ugroupe). Average quare evato from the mea: : = =. Average quare evato from the arthmetc average (Varace) (populato varace) σ : = µ N = ( ) : = = (ample varace) 3. Staar evato σ : = σ, σ > 0 (populato taar evato) : =, > 0 (populato taar evato) 3

4 E. 3.. (cot.) ( ) = : = 6.0 = = ( ) : = 6. = : =.4 R (Short-Cut Formula for the Varace for Ugroupe Data) σ = = = N N = = = D (Average Square Devato - Groupe) : 0.5 p = 0.5 F p σ : µ F N (populato varace) = p : F = (ample varace) R I cae the formato E. 3.. (cot.) We ha: are ot avalable, they wll be replace by the terval mpot.: 0.5 = 6.9, =

5 C F m m 0.5 F m [ 4.00, 5.00[ [ 5.00, 6.00[ [ [ [ [ Total F , c =.4,.07 9 R (Short-Cut Formula for the Varace for Groupe Data) σ = = F N = F N (populato varace) = = F = F (ample varace) T. 4.. (Chebyhev) For ay umber the mea. k >, at leat ( k ) of the ata value le wth k taar evato of E. 4.. The arthmetc mea bweekly amout cotrbute by the employee of a compay to compay proft-harg pla $5.54, a the taar evato $7.5. At leat what percet of the cotrbuto le wth plu 3.5 taar evato a mu taar evato of the mea? Soluto: = = 0.9. k ( 3.5) 5

6 R (Emprcal Rule, Three-Sgma Rule) Wherea Chebyhev theorem applcable to ay k of trbuto, the emprcal rule apple oly to a pecfc type of trbuto calle a ormal trbuto: For a ormal trbuto, appromately. 68% of the obervato le wth oe taar evato of the mea. 95% of the obervato le wth two taar evato of the mea % of the obervato le wth three taar evato of the mea R At leat 50% of the obervato a ata et le the terval [ µ σ, µ + σ ], (populato), + (ample) D (Coeffcet of Varato) σ v : = µ (populato) v : = (ample) R The coeffcet of varato ue to compare the varablty of ata et wth fferet arthmetc average. E. 3.. (cot.). Ugroupe:. Groupe:.4 v : = v : = R (A Rule of Thumb) The arthmetc average of a ata et to be coere a repreetatve oly f t coeffcet of varato le tha 0.5 (or 50%). D (Skewe) The coeffcet of kewe efe a follow: 6

7 3 Me S : =, ( 3 S + 3) E Ugroupe ( ) S = St. Dev =,4 Mea = 6, 0 N =,00 4,00 5,00 6,00 7,00 8,00 4,50 5,50 6,50 7,50 8,50 wage per hour. Groupe ( ) S 0.9. D (Bo-a-Whker Plot) A bo-a-whker Plot a plot that how the cetre, prea, a kewe of a ata et. R It cotructe by rawg a bo a two whker that ue the mea, the frt (lower) quartle, the thr (upper) quartle, a the mallet a the larget value the ata et betwee the lower a the upper er fece. The followg eample epla all the tep eee to make a bo-a-whker plot E. 4.. The followg ata are the come [ thoua of ] for a ample of houehol:

8 Cotruct a bo-a-whker plot of thee ata. Soluto: Step F the value of the mea, the frt quartle, the thr quartle, a the terquartle rage: R Q = 47, = 37, = 6, = 6 37 = 4. Step F the pot that are.5 RQ below 0.5 a.5 RQ above Thee two pot are calle the lower a the upper er fece, repectvely:.5 R Q =.5 4 = 36 Lower er fece = = = Upper er fece = = = 97. Step 3 Determe the mallet a the larget value the gve ata et wth the two er fece: Smallet value wth the two er fece = 9 Larget value wth the two er fece = 7 Step 4 Draw a horzotal le a mark the come level o t uch that all the value the gve ata et are covere. Above the horzotal le, raw a bow wth t left e at the poto of the frt quartle a the rght e at the poto of the thr quartle. Ie the bo, raw a vertcal le at the poto of the mea: The reult how the followg fgure: Step 5 By rawg two le, jo the pot of the mallet a the larget value wth the two er fece to the bo. Thee value are 9 a 7 th eample. The two le that jo the bo to thee two value are calle whker. A value that fall oute the two er fece how by makg a aterk a calle a outler. Th complete the bo-a-whker plot, a how the followg fgure: 8

9 I the above fgure, about 50% of the ata value fall wth the bo, about 5% of the value fall o the left e of the bo, a about 5% fall o the rght e of the bo. Alo, 50% of the value fall o the left e of the mea a 50% le o the rght e of the mea. The ata of th eample are kewe to the rght becaue the lower 50% of the value are prea over a maller rage tha the upper 50% of the value. R. 4.. The obervato that fall oute the two er fece are calle outler. Thee outler ca be clafe to two k: ml a etreme outler. To o o we efe two outer fece a lower outer fece at 3.0 before the frt quartle a a upper outer fece at 3.0 above the thr quartle. If a obervato oute ether RQ of the two er fece but wth ether of the two outer fece, t calle a ml outler. A obervato that oute ether of the two outer fece, calle a etreme outler. For the above eample, the outer fece are -35 a 33. Becaue 44 oute the upper er fece but e the upper outer fece, t a ml outler. For a ymmetrc ata et, the le repreetg the mea wll be the mle of the bo a the prea of the value wll be over almot the ame rage o both e of the bo. E. 3.. (cot.) Bo-a-Whker Plot: RQ N = w age per hour 9

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- Inferential: methods using sample results to infer conclusions about a larger pop n.

- Inferential: methods using sample results to infer conclusions about a larger pop n. Chapter 6 Def : Statstcs: are commoly kow as umercal facts. s a feld of dscple or study. I ths class, statstcs s the scece of collectg, aalyzg, ad drawg coclusos from data. The methods help descrbe ad